Q69 / 180 · COMEDK 2021 · Mathematics
If $A(\operatorname{adj} A)=\left[\begin{array}{ccc}-2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2\end{array}\right]$, then $|\operatorname{adj} A|$ equals
- A$-2$
- B$-4$
- Correct answerC4
- D8
Solution
Given $A(\operatorname{adj} A) = \begin{bmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{bmatrix} = -2I$, where $I$ is the $3 \times 3$ identity matrix.
Using the property $A(\operatorname{adj} A) = |A|I$, we have $|A| = -2$.
The determinant of the adjoint of a matrix $A$ of order $n$ is given by $|\operatorname{adj} A| = |A|^{n-1}$.
Here, $n = 3$ and $|A| = -2$.
Therefore, $|\operatorname{adj} A| = (-2)^{3-1} = (-2)^2 = 4$.
Answer: 4
Using the property $A(\operatorname{adj} A) = |A|I$, we have $|A| = -2$.
The determinant of the adjoint of a matrix $A$ of order $n$ is given by $|\operatorname{adj} A| = |A|^{n-1}$.
Here, $n = 3$ and $|A| = -2$.
Therefore, $|\operatorname{adj} A| = (-2)^{3-1} = (-2)^2 = 4$.
Answer: 4
Hence, the correct answer is option: 'C'.